# Properties

 Label 1369.2.b.b Level $1369$ Weight $2$ Character orbit 1369.b Analytic conductor $10.932$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1369 = 37^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1369.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.9315200367$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 37) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} -i q^{5} + 2 q^{7} + 3 i q^{8} -3 q^{9} +O(q^{10})$$ $$q + i q^{2} + q^{4} -i q^{5} + 2 q^{7} + 3 i q^{8} -3 q^{9} + q^{10} + 2 q^{11} + 2 i q^{13} + 2 i q^{14} - q^{16} -3 i q^{17} -3 i q^{18} + 6 i q^{19} -i q^{20} + 2 i q^{22} + 4 i q^{23} + 4 q^{25} -2 q^{26} + 2 q^{28} + 9 i q^{29} -10 i q^{31} + 5 i q^{32} + 3 q^{34} -2 i q^{35} -3 q^{36} -6 q^{38} + 3 q^{40} + 9 q^{41} -2 i q^{43} + 2 q^{44} + 3 i q^{45} -4 q^{46} + 6 q^{47} -3 q^{49} + 4 i q^{50} + 2 i q^{52} -2 q^{53} -2 i q^{55} + 6 i q^{56} -9 q^{58} + 4 i q^{59} + i q^{61} + 10 q^{62} -6 q^{63} -7 q^{64} + 2 q^{65} + 10 q^{67} -3 i q^{68} + 2 q^{70} + 6 q^{71} -9 i q^{72} + 10 q^{73} + 6 i q^{76} + 4 q^{77} -10 i q^{79} + i q^{80} + 9 q^{81} + 9 i q^{82} -12 q^{83} -3 q^{85} + 2 q^{86} + 6 i q^{88} + 7 i q^{89} -3 q^{90} + 4 i q^{91} + 4 i q^{92} + 6 i q^{94} + 6 q^{95} -7 i q^{97} -3 i q^{98} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q + 2q^{4} + 4q^{7} - 6q^{9} + 2q^{10} + 4q^{11} - 2q^{16} + 8q^{25} - 4q^{26} + 4q^{28} + 6q^{34} - 6q^{36} - 12q^{38} + 6q^{40} + 18q^{41} + 4q^{44} - 8q^{46} + 12q^{47} - 6q^{49} - 4q^{53} - 18q^{58} + 20q^{62} - 12q^{63} - 14q^{64} + 4q^{65} + 20q^{67} + 4q^{70} + 12q^{71} + 20q^{73} + 8q^{77} + 18q^{81} - 24q^{83} - 6q^{85} + 4q^{86} - 6q^{90} + 12q^{95} - 12q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1369\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1368.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 1.00000i 0 2.00000 3.00000i −3.00000 1.00000
1368.2 1.00000i 0 1.00000 1.00000i 0 2.00000 3.00000i −3.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1369.2.b.b 2
37.b even 2 1 inner 1369.2.b.b 2
37.d odd 4 1 1369.2.a.b 1
37.d odd 4 1 1369.2.a.d 1
37.g odd 12 2 37.2.c.a 2
111.m even 12 2 333.2.f.a 2
148.l even 12 2 592.2.i.c 2
185.p even 12 2 925.2.o.a 4
185.q odd 12 2 925.2.e.a 2
185.u even 12 2 925.2.o.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.c.a 2 37.g odd 12 2
333.2.f.a 2 111.m even 12 2
592.2.i.c 2 148.l even 12 2
925.2.e.a 2 185.q odd 12 2
925.2.o.a 4 185.p even 12 2
925.2.o.a 4 185.u even 12 2
1369.2.a.b 1 37.d odd 4 1
1369.2.a.d 1 37.d odd 4 1
1369.2.b.b 2 1.a even 1 1 trivial
1369.2.b.b 2 37.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1369, [\chi])$$:

 $$T_{2}^{2} + 1$$ $$T_{3}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$81 + T^{2}$$
$31$ $$100 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$( -9 + T )^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$( -6 + T )^{2}$$
$53$ $$( 2 + T )^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$1 + T^{2}$$
$67$ $$( -10 + T )^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$100 + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$49 + T^{2}$$
$97$ $$49 + T^{2}$$
show more
show less